On the rigidity theorem for harmonic functions in Khler metric of Bergman type

The paper gives a method to generate the potential functions which can induce Khler metrics u = uij dz idz j of Bergman type on the unit ball B n in C n . The paper proves that if h ∈ C n (B n ) is harmonic in these metrics u ( u h = 0) in B n , then h must be pluriharmonic in B n . In fact, it is a characterization theorem, as a consequence, the paper provides a way to construct many counter examples for the potential functions of the metric u so that the above theorem fails. The results in this paper generalize the theorems of Graham (1983) and examples constructed by Graham and Lee (1988).